We have found one software suitable in our database for this conversion.
You can save .aww documents to .docx files directly Ability Write, which is essentially what aww to docx conversion refers to. Unfortunately, .aww files are exclusive to Ability Write and no 3rd party word processor seems to support this format.
Also worth mentioning is that as DOCX is a proprietary format of Microsoft Word, there might be some metadata and special formatting missing or not properly converted. The text body however, should be intact.
AWW
Files with aww file extension can be most notably found as documents made using Ability Write word processor, part of Ability Office suite.
DOCX
Files with docx extension are documents written and saved using Microsoft Word word processor edition 2007 or later. DOCX format replaced the previously used DOC format, moving from binary files to Open XML schema-based documents. They are supported in almost every modern word-processing tools.